Stochastic Preconditioning for Neural Field Optimization

TL;DR

This work introduces stochastic preconditioning (SP), a simple technique that perturbs query locations with Gaussian noise to implicitly blur neural fields during optimization. Seen as an implicit blur operator, SP improves convergence, reduces artifacts, and enhances robustness across a variety of neural-field representations (e.g., SDFs, hashgrid, triplane, NeRF variants) and tasks (surface reconstruction, radiance field estimation), often matching or outperforming task-specific hierarchies with negligible overhead. The method uses a single-sample estimate of the blurred field, with an annealing schedule for the blur radius $\alpha$ and optional spatially varying maps $\alpha(\mathbf{x})$, making it easy to integrate into existing codebases. SP demonstrates consistent gains in indirectly supervised settings and provides immediate improvements in directly supervised cases, offering a unified approach to improve neural-field optimization with broad practical impact.

Abstract

Neural fields are a highly effective representation across visual computing. This work observes that fitting these fields is greatly improved by incorporating spatial stochasticity during training, and that this simple technique can replace or even outperform custom-designed hierarchies and frequency space constructions. The approach is formalized as implicitly operating on a blurred version of the field, evaluated in-expectation by sampling with Gaussian-distributed offsets. Querying the blurred field during optimization greatly improves convergence and robustness, akin to the role of preconditioners in numerical linear algebra. This implicit, sampling-based perspective fits naturally into the neural field paradigm, comes at no additional cost, and is extremely simple to implement. We describe the basic theory of this technique, including details such as handling boundary conditions, and extending to a spatially-varying blur. Experiments demonstrate this approach on representations including coordinate MLPs, neural hashgrids, triplanes, and more, across tasks including surface reconstruction and radiance fields. In settings where custom-designed hierarchies have already been developed, stochastic preconditioning nearly matches or improves their performance with a simple and unified approach; in settings without existing hierarchies it provides an immediate boost to quality and robustness.

Figures (13)

• Figure 1: Stochastic preconditioning can easily be applied to a wide variety of existing neural field representations, consistently reducing artifacts and improving quality, here in the context of fitting signed-distance functions to oriented point clouds.
• Figure 2: Example integration of stochastic preconditioning in an existing implementation of a neural field-based method.
• Figure 3: A simple example of fitting stochastically-blurred neural fields to 1D (top) and 2D (bottom) signals. Sampling from the blurred $f(\bm{x} + \bm{\delta})$ while fitting yields a field $f(\bm{x})$ which is itself blurred. The level of blur closely approximates the corresponding blur of the true signal (top, inset).
• Figure 4: Boundary handling. Stochastic preconditioning will perturb samples outside of bounded domains. Clamping would incorrectly concentrate samples on the boundary, while reflecting around the boundary retains the expected uniform distribution.
• Figure 5: SDFs from oriented point clouds. Stochastic preconditioning reduces artifacts and improves quality when fitting SDFs to point clouds, here using an INGP Hashgrid MLP instantngp as the field representation, along with geometric initialization sal. Here we visualize Chamfer error against known ground-truth. The standard approach exhibits artifacts which are not resolved by training, while stochastic preconditioning successfully converges to a high-fidelity surface.